Chapter 1.1, Problem 61AYU

Baseball Refer to Problem $63$. Overlay a rectangular coordinate system on a major league baseball diamond so that the origin is at home plate, the positive $x$-axis lies in the direction from home plate to first base, and the positive $y$-axis lies in the direction from home plate to third base.

What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement.

If the right fielder is located at $\left(310,15\right)$ how far is it from the right fielder to second base?

If the center fielder is located at $\left(300,300\right)$, how far is it from the center fielder to third base?

To determine

The coordinate of first base, second base and third base in the feet as unit of measurement.

Answer to Problem 61AYU

Solution:

The coordinate of first base is $\left(90,0\right)$, second base is $\left(90,90\right)$, and the third base is $\left(0,90\right)$

Explanation of Solution

Given Information:

A major league baseball “diamond” is square with $90$ feet on a side.

A rectangular co-ordinate system on a major league baseball is diamond. The origin is at home plate, the positive $x-$ axis lies in the direction from home plate to first base, and the positive $y-$ axis lies in the direction from home plate to third base as shown in below figure:

Explanation:

As the home plate is at the origin of the rectangular coordinate then the coordinate of the home plate is origin that means $\left(0,0\right)$

The first base is at a distance of $90$ feet from the home plate along the positive $x-$ axis, so the coordinate of the first base is $\left(90,0\right)$

The second base is at a distance of $90$ feet from the first base along the positive $y-$ axis, so the coordinate of the second base is $\left(90,90\right)$

The third base is at a distance of $90$ feet from the home plate along the positive $y-$ axis, so the coordinate of the second base is $\left(0,90\right)$

The below figure shows all the vertices of the baseball diamond:

To determine

To calculate: The distance from the right fielder to second base where the right fielder is located at $\left(310,15\right)$

Answer to Problem 61AYU

Solution:

The distance between the right fielder to second base is $232.43\text{ft}$

Explanation of Solution

Given Information:

A major league baseball “diamond” is square $90$ feet on a side.

A rectangular co-ordinate system on a major league baseball is diamond. The origin is at home plate, the positive $x-$ axis lies in the direction from home plate to first base, and the positive $y-$ axis lies in the direction from home plate to third base as shown in below figure:

Formula used:

The distance formula: The distance between two points $P_1=(x_1,y_1)$ and $P_2=(x_2,y_2)$ denoted by $d\left(P_1,P_2\right)$, is $d\left(P_1,P_2\right)=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$.

Calculation:

From figure and part (a),

Thesecond base located at the $\left(90,90\right)$

The right fielder is located at the $\left(310,15\right)$

The distance between the right fielder to second base is $d$

By using the distance formula

$d=\sqrt{{\left(310-90\right)}^2+{\left(15-90\right)}^2}$

$d=\sqrt{{\left(220\right)}^2+{\left(-75\right)}^2}$

$d=\sqrt{48400+5625}$

$d=\sqrt{54025}$

$d=5\sqrt{2161}$

$d\approx 232.43\text{ft}$

Therefore, the distance between the right fielder to second base is approximately $232.43\text{ft}$

To determine

To calculate: The distance from the center fielder to third base, where the center fielder is located at $\left(300,300\right)$.

Answer to Problem 61AYU

Solution:

The distance between the center fielder to third base is approximately $366.20\text{ft}$

Explanation of Solution

Given Information:

A major league baseball “diamond” is square $90$ feet on a side.

A rectangular co-ordinate system on a major league baseball is diamond. The origin is at home plate, the positive $x-$ axis lies in the direction from home plate to first base, and the positive $y-$ axis lies in the direction from home plate to third base as shown in below figure:

Formula used:

The distance formula: The distance between two points $P_1=(x_1,y_1)$ and $P_2=(x_2,y_2)$ denoted by $d\left(P_1,P_2\right)$, is $d\left(P_1,P_2\right)=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

Calculation:

From figure and part (a),

Thethird base located at the $\left(0,90\right)$

The center fielder is located at $\left(300,300\right)$

The distance between the center fielder to third base is $d$

By using the distance formula

$d=\sqrt{{\left(300-0\right)}^2+{\left(300-90\right)}^2}$

$d=\sqrt{{\left(300\right)}^2+{\left(210\right)}^2}$

$d=\sqrt{90000+44100}$

$d=\sqrt{134100}$

$d=30\sqrt{149}\text{ft}$

$d\approx 366.20\text{ft}$

Therefore, the distance between the center fielder to third base is approximately $366.20\text{ft}$